Aztec Information

Standard Western astrology is based upon the motion of the planets in the sky, the realm of physics. In contrast, the astrology of the Aztecs is based on repeating sequences of numbers, the stuff of number theory. As a mathematician, I hope to be a little subversive in promoting this latter approach!

This page then contains some information about the Aztec calendar and some bits of the mathematics which are useful in using it. Much of it is motivated by questions I've been asked - the remainder is very much "under construction".

This page is best viewed with a subscript-capable browser, such as Netscape 2.0 or better.

The Astrological Calendar

The astrological calendar consists of 20 days which repeat themselves over and over.
  1. Cipactli
  2. Ehecatl
  3. Calli
  4. Cuetzpallin
  5. Coatl
  6. Miquiztli
  7. Mazatl
  8. Tochtli
  9. Atl
  10. Itzcuintli
  11. Ozomatli
  12. Malinalli
  13. Acatl
  14. Ocelotl
  15. Cuauhtli
  16. Cozcaquauhtli
  17. Ollin
  18. Tecpatl
  19. Quiahuitl
  20. Xochitl
Each day is additionally numbered from 1 to 13, again a repeating sequence. Since 20 and 13 are different, the days in the second cycle will be numbered differently to those in the first. These two sequences will only come into line every 260 days (since 260 is the least common multiple of 20 and 13).

The 13-day period gives the Aztec week. Each with starts when a day is numbered 1, with the week named by the name of that day.

The Chinese Remainder Theorem

What does an old Chinese theorem have to do with the Aztec calendar? Well, suppose we want to find a day in the Gregorian calendar corresponding to a given day in the 260-day astrological calendar. It is straightforward to make a list of all 260 days and to look up the one we want. Alternatively, we can use the following famous theorem to directly calculate which day it is.

Chinese Remainder Theorem. Let m1, ..., mr be pairwise relatively prime positive integers. Then the system of equations

x = ai mod mi,

where i = 1,...,r, for given integers ai, has a unique solution modulo m = m1m2...mr. Furthermore, the solution is given by
x = a1M1y1+ ... + arMryr ,

where Mi = m/mi and yi is such that Miyi = 1 mod mi (i.e. yi is the inverse of Mi mod mi).

Example. Suppose we want to find the next occurrence of 5 Itzcuintli. Firstly, in modulo arithmetic numbering traditionally begins with 0, so we will treat the 5 as number 4. Itzcuintli is the 10th day in the calendar, which we have as number 9. Thus, we want the day, x say, in the 260-day calendar such that x is the 5th day in the 13-day numbering cycle and the 10th day in the 20-day cycle of days. As equations, we want x such that

x = 4 mod 13
x = 9 mod 20

From the theorem above, m1 = 13, m2 = 20, m = 13.20 = 260, M1 = 260/13 = 20, M2 = 260/20 = 13, a1 = 4, a2 = 9. To calculate the yi is another story - we find y1 = 2 and y2 = 17. This gives

x = 4.20.2 + 9.13.17 = 69 mod 260.

Thus 5 Itzcuintli is the 70th day in the 260 day period.

Julian Days

The Julian Calendar was introduced by Julius Caesar in 46 B.C. (and later modified under Augustus), giving a system in which an extra day was added to the 365-day year every four years. (Over 52 years this is 13 days, equivalent then to the Aztec solar calendar).

Another calendar system also bears the name "Julian", but is much more recent and is named for a different Julius. Joseph Scaliger of Leyden introduced the system of julian days (named after his father) in 1583 as a method of time-keeping which was independent of historical calendars. Day 1 in the system is 1 January 4713 B.C., with Julian days counted from then on. For example, 11 September 1996 is Julian Day 2450338.

Julian days provide a standard for converting between calendars, such as between the Gregorian and Aztec calendars.

Details to come...

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